An internal definition in a realizability topos

[Andrej Bauer <andrej.bauer@andrej.com>]

Consider a realizability topos RT(A) over some PCA A. There is an
embedding Nabla : Set -> RT(A), which takes a set X to the assembly
whose underlying set is X and the existence predicate is trivial,
i.e., every element of X is realized by every realizer of A.

Let [n] = {0, 1, 2, .., n-1} be the set with n elements. I have some
interest in the subobject of Nabla([n]) whose underlying set is [n]
and the existence predicate is

   E(0) = { numeral(0) }
   E(1) = { numeral(0), numeral(1) }
   ...
   E(k) = { numeral(k-1), numeral(k) }
   ...
   E(n-1) = {numeral(n-2)}

In words: there are n elements 0, 1, 2, ..., n-1, where two
consequtive elements share a realizer.

The question is; is this object definable in the internal language of the topos?

Note that Nabla(2) is definable as the object of not-not-stable truth
values, and that each Nabla(X) is also definable with a bit more work.

With kind regards,

Andrej


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